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A longstanding problem in natural science and later in physics was the understanding of the existence of ferromagnetism and its disappearance under heating to high temperatures. Although a qualitative description was possible by the Curie–Weiss theory, it was obvious that a microscopic model was necessary to explain the tendency of the elementary magnetons to prefer parallel ordering at low temperatures. Such a model was proposed in 1922 by Schottky within the old Bohr–Sommerfeld quantum mechanics and claimed to explain the high values of the Curie temperatures of certain ferromagnets. Based on this idea Ising formulated a new model for ferromagnetism in solids. Simultaneously the old quantum mechanics was replaced by new concepts of Heisenberg and Schrödinger and the discovery of spin. Thus Schottky’s idea was outperformed and finally replaced in 1928 by Heisenberg exchange interaction. This led to a reformulation of Ising’s model by Pauli at the Solvay conference in 1930. Nevertheless one might consider Schottky’s idea as a forerunner of this development explaining and asserting that the main point is the Coulomb energy leading to the essential interaction of neighboring elementary magnets.

On this, the occasion of the 20th anniversary of the "Ising Lectures" in Lviv (Ukraine), we give some personal reflections about the famous model that was suggested by Wilhelm Lenz for ferromagnetism in 1920 and solved in one dimension by his PhD student, Ernst Ising, in 1924. That work of Lenz and Ising marked the start of a scientific direction that, over nearly 100 years, delivered extraordinary successes in explaining collective behaviour in a vast variety of systems, both within and beyond the natural sciences. The broadness of the appeal of the Ising model is reflected in the variety of talks presented at the Ising lectures ( http://www.icmp.lviv.ua/ising/ ) over the past two decades but requires that we restrict this report to a small selection of topics. The paper starts with some personal memoirs of Thomas Ising (Ernst's son). We then discuss the history of the model, exact solutions, experimental realisations, and its extension to other fields., 46 pages, 9 figures

We calculate in two loop order the dynamical critical behavior at the tricritical point occuring in 3He-4He mixtures. Model F′ introduced by Siggia and Nelson (Phys. Rev. B 15:1427, [1977])—already problematic in one loop order—is shown in two loop order to lead to a divergence of the mass diffusion D in contradiction to experiments. Within the complex symmetric version of this model (model E⋆′) the dynamical critical exponents of the kinetic coefficients can be calculated exactly leading to a mass diffusion going to zero at the tricritical point as D(tX)∼tX1/2 (tX is the relative temperature distance at constant concentration) and the thermal diffusion ratio kT∼tX−1 diverges proportional to the concentration susceptibility in the experimental region. We also discuss effective dynamical critical exponents and compare them with recent simulations of the vector Blume-Emery-Griffths model.

The scaling behavior of model C describing the dynamical behaviour of the $n$-component nonconserved order parameter coupled statically to a scalar conserved density is considered in $d$-dimensional space. Conditions for the realization of different types of scaling regimes in the $(n,d)$ plane are studied within the field-theoretical renormalization group approach. Borders separating these regions are calculated on the base of high-order RG functions using $\epsilon$-expansions as well as by fixed dimension $d$ approach with resummation., Comment: 8 pages, 2 figures

We study the relaxational critical dynamics of the three-dimensional random anisotropy magnets with the non-conserved n-component order parameter coupled to a conserved scalar density. In the random anisotropy magnets the structural disorder is present in a form of local quenched anisotropy axes of random orientation. When the anisotropy axes are randomly distributed along the edges of the n-dimensional hypercube, asymptotical dynamical critical properties coincide with those of the random-site Ising model. However structural disorder gives rise to considerable effects for non-asymptotic critical dynamics. We investigate this phenomenon by a field-theoretical renormalization group analysis in the two-loop order. We study critical slowing down and obtain quantitative estimates for the effective and asymptotic critical exponents of the order parameter and scalar density. The results predict complex scenarios for the effective critical exponent approaching an asymptotic regime., Comment: 8 figures, style files included

The critical dynamics of model C in the presence of disorder is considered. It is known that in the asymptotics a conserved secondary density decouples from the nonconserved order parameter for disordered systems. However couplings between order parameter and secondary density cause considerable effects on non-asymptotic critical properties. Here, a general procedure for a renormalization group treatment is proposed. Already the one-loop approximation gives a qualitatively correct picture of the diluted model C dynamical criticality. A more quantitative description is achieved using two-loop approximation. In order to get reliable results resummation technique has to be applied., 21 page, 4 figures, submitted to the proceedings of the conference "Renormalization Group 2005", Helsinki, Finland, 30 August - 3 September 2005

We review the progress made in dynamic bulk critical behaviour in equilibrium in the last 25 years since the review of Halperin and Hohenberg. We unify the presentation of the theoretical background by restricting ourselves to the field-theoretic renormalization group method. The main results obtained in the different universality classes are presented. This contains the critical dynamics near the gas–liquid transition in pure fluids (model H), the plait point and consolute point in mixtures (model H � ), the superfluid transition in 4 He (model F) and 4 He– 3 He mixtures (model F � ), the Curie point (model J) and Neel point (model G) in Heisenberg magnets and the superconducting transition. In comparison with experimental results, it became clear that in most cases one has to consider apart from the universal asymptotic critical behaviour also the non-universal effective behaviour. Either because it turned out to be inevitable due to a small dynamical transient exponent inhibiting the system to reach the asymptotics (e.g., at the superfluid transition) or because one is interested in the region further away from the phase transition like in pure fluids and mixtures at their gas–liquid or demixing transition. The calculation of the critical dynamics is adequate in most cases only in two-loop order. We review these results and present the solution to unreasonable features found for some models. Thus, we consider model C where relaxational and diffusive dynamics are coupled and the scaling properties and the limit to a purely relaxational model (model A) have not been understood. In general for models where the order parameter couples to other conserved densities time scale ratios between the kinetic coefficients of the order parameter and the conserved densities play an important role. Their fixed-point values and the approach to the fixed point are changed considerably in two-loop order compared to their values in one-loop order. These considerations are relevant for the explanation of the dynamical critical shape functions of systems such as superfluid helium (model F) and the isotropic antiferromagnet (model G). As far as possible, the comparison of results obtained by the renormalization group theory with numerical simulations has been made.

An integral equation approach is developed to investigate phase coexistence properties of Ising spin fluids with Yukawa ferromagnetic and Lennard-Jones nonmagnetic interactions in the presence of an external field. The calculations are carried out on the basis of the Duh and Henderson closure with a specific Duh-like partitioning of the total potential. The coupled set of the Ornstein-Zernike equation, the closure relation and the external field constraint are solved using an efficient numerical algorithm. The phase diagrams are evaluated in a wide range of varying the external field and the ratio of strengths of Yukawa to Lennard-Jones interactions. Different types of the phase diagram topology as well as various external field dependencies of critical temperatures and densities are identified. The complexity with respect to simple Lennard-Jones fluids is explained by coupling between spatial and spin degrees of freedom in the system. A comparison of the obtained theoretical results with simulation data is made and a good agreement is observed.

The problem of critical behaviour of three dimensional random anisotropy magnets, which constitute a wide class of disordered magnets is considered. Previous results obtained in experiments, by Monte Carlo simulations and within different theoretical approaches give evidence for a second order phase transition for anisotropic distributions of the local anisotropy axes, while for the case of isotropic distribution such transition is absent. This outcome is described by renormalization group in its field theoretical variant on the basis of the random anisotropy model. Considerable attention is paid to the investigation of the effective critical behaviour which explains the observation of different behaviour in the same universality class., Comment: 41 pages, 10 figures

Universal and nonuniversal critical exponents of a three-dimensional Ising system with weak quenched disorder are discussed. Experimental, computational, and theoretical results are reviewed. Particular attention is given to field-theoretical renormalization-group results. Different renormalization schemes are considered with emphasis on the analysis of the divergent series obtained.

The main aims of this paper are: (i) to present molecular dynamics (MD) results for time correlation functions, calculated with the help of a new symplectic algorithm that was proposed by us recently; (ii) to discuss the results obtained with the main focus on the interplay between the spin and liquid subsystems as well as on the influence of an external magnetic field on the properties of the system considered; (iii) to compare the time correlation functions, calculated in MD simulations, with theoretical predictions. Our results provide evidence of the interesting interplay of the two subsystems, causing new phenomena not found in nonmagnetic fluids and magnetic lattice models.

In these lectures, we discuss the influence of weak quenched disorder on the critical behavior in condensed matter and give a brief review of available experimental and theoretical results as well as results of MC simulations of these phenomena. We concentrate on three cases: (i) uncorrelated random-site disorder, (ii) long-range-correlated random-site disorder, and (iii) random anisotropy. Today, the standard analytical description of critical behavior is given by renormalization group results refined by resummation of the perturbation theory series. The convergence properties of the series are unknown for most disordered models. The main object of these lectures is to discuss the peculiarities of the application of resummation techniques to perturbation theory series of disordered models., Lectures given at the Second International Pamporovo Workshop on Cooperative Phenomena in Condensed Matter (28th July - 7th August 2001, Pamporovo, Bulgaria). 51 pages, 12 figures, 1 style files included

Recent low-gravity measurements of the shear viscosity in xenon near its critical point are compared with theoretical results obtained within the field theoretical renormalization group (RG) theory. Nonasymptotic effects and gravity effects are included in our theoretical description, which allows a comparison outside the asymptotic region as well as with earthbound experiments affected by gravity. Comparisons are also made with the theoretical result of the mode coupling theory. In both theories no agreement with the frequency dependence of the real part of the shear viscosity within one-loop theory can be reached. The experimental value of the ratio of the imaginary part to the real part of the shear viscosity at Tc is found to be in agreement with the value calculated within the decoupled mode theory using the two-loop value for the critical exponent of the temperature dependence of the shear viscosity, but not with the one-loop value obtained in RG theory. Thus a complete two-loop calculation of the vertex function for the shear viscosity is required.

Six-loop massive scheme renormalization group functions of a d=3-dimensional cubic model (J.M. Carmona, A. Pelissetto, and E. Vicari, Phys. Rev. B vol. 61, 15136 (2000)) are reconsidered by means of the pseudo-epsilon expansion. The marginal order parameter components number N_c=2.862(5) as well as critical exponents of the cubic model are obtained. Our estimate N_c, 8 pages

The leading correction-to-scaling exponent $\omega$ for the three-dimensional dilute Ising model is calculated in the framework of the field theoretic renormalization group approach. Both in the minimal subtraction scheme as well as in the massive field theory (resummed four loop expansion) excellent agreement with recent Monte Carlo calculations [Ballesteros H G, et al Phys. Rev. B 58, 2740 (1998)] is achieved. The expression of $\omega$ as series in a $\sqrt{\epsilon}$-expansion up to ${\cal O}(\epsilon^2)$ does not allow a reliable estimate for $d=3$., Comment: 4 pages, latex, 1 eps-figure included

Within mean field approximation we investigate the phase diagrams of magnetic fluids in presence of a magnetic field. In a finite field the magnetic phase transition is absent, but instead a line of first order liquid-liquid transitions ending in a critical point occurs for a magnetic interaction, which is sufficiently strong. Varying the magnetic field these critical points extend from the tricritical point at H=0 to a critical endpoint. For a fluid with Ising spins we calculate the critical lines and several tricritical exponents analytically. For Heisenberg fluids we obtain the phase diagrams from a numerical solution of the mean field equations of state.

Solid state systems exhibit besides usual second order phase transitions a rich variety of multicritical phenomena like Lifshitz points (or lines), tricritical points (or lines) and even tricritical Lifshitz points. Realizations of such points are numerous and were also verified in the family of ferroelectrics of the type (PbySn1y)2P2(SexS1-x)6. A review of the critical behavior at such points is presented here. Because of the importance of the uniaxial dipolar interaction in ferroelectrics the critical behavior is different from systems with short range interaction only. Moreover the coupling to the elastic degrees of freedom may not be neglected, and leads under certain conditions to a critical temperature dependence in certain elastic constants. Crossover phenomena, which are expected in the experimental accessible region of experiments are also considered.

We derive the nonasymptotic expressions for the frequency- and temperature-dependent sound velocity and sound absorption near a critical point in a mixture within renormalization group theory in one-loop order. The dynamic model considered is an extension of the corresponding model for pure fluids including concentration fluctuations. The theoretical result for the complex sound velocity is the same as at consolute points and gas-liquid critical points reflecting universality. Differences observed in the experiments at the two critical points mentioned are due to the different behavior of the sound velocity at ${T}_{c},$ which is finite in mixtures and zero in pure fluids, as well as due to nonasymptotic effects. Near the consolute point we compare our result with the phenomenological theory of Ferrell and Bhattacharjee [Phys. Rev. B 24, 4095 (1981); Phys. Rev. A 31, 1788 (1985)] and near the gas-liquid critical point with experiments in the ${}^{3}{\mathrm{H}\mathrm{e}\ensuremath{-}}^{4}\mathrm{He}$ mixture. A genuine dynamic parameter not considered so far and related to the critical enhancement of the thermal conductivity appears in the nonasymptotic expressions of the transport coefficients and the complex sound velocity. All nonuniversal background parameters of the complex sound velocity are fixed by a comparison of the corresponding theoretical expressions for the transport coefficients with experiments.

We compare the theoretical results of an explicit one loop calculation of the critical behavior of the sound propagation in pure liquids near the gas-liquid critical point, which has been derived within the field-theoretic renormalization group formalism, with experimental data in ${}^{3}$He, ${}^{4}$He, CO${}_{2}$, SF${}_{6}$, and Xe. The nonuniversal initial values of two dynamic model parameters, which are necessary for the calculation of all theoretical expressions, are determined by a fit of the shear viscosity at zero frequency in a small temperature region. The static quantities appearing in the theoretical expressions are taken from experiment. With these two dynamical initial values the temperature flow of the dynamic model parameters is completely determined. The sound attenuation and the sound velocity at arbitrary frequency as well as the thermal conductivity or the thermal diffusion coefficient may be calculated without any adjustable parameter. The parameter free predictions are in very good agreement with experimental results. This also holds for scaling plots of the reduced attenuation and dispersion taking into account the nonasymptotic behavior of the dynamic scale.

We calculate the hydrodynamic shear and sound mode near the liquid-gas phase transition and predict its values for different frequencies and temperatures. The dynamical parameters that enter the theoretical expressions are apart from static quantities the background values of the Onsager coefficients for the order parameter and the transverse momentum current. We find within the field-theoretical renormalization group formalism the asymptotic scaling functions or the real and imaginary parts of the shear and sound mode in the $\ensuremath{\omega}$-${\ensuremath{\xi}}^{\ensuremath{-}1}$ plane. Our main concern, however, is the calculation of the nonasymptotic expressions describing the crossover from the analytic background to the asymptotic critial behavior.

We study the recognition and correction properties of a structured associative memory with a floating threshold which can be considered a modification of the Willshaw network and bears some resemblance to a network of Potts neurons. An analytical study is performed for the asymptotic case of large networks and single-step retrieval. The main informational characteristics of the network are obtained; the number of spurious stable states and the size of their attraction basins are calculated. A comparison is made with the Willshaw network considered earlier in this series of papers.

On the basis of our previous results (Physica A 220 (1995) 325) the collective mode spectrum of the Heisenberg ferrofluid model is studied in the hydrodynamic region. We start from the rigorous microscopic approach taking into account all conserved quantities and calculate the explicit expressions for the dispersion and damping coefficient of sound modes. It is shown that the sound velocity of Heisenberg ferrofluid at constant external magnetic field is isotropic and can be expressed by the adiabatic compressibility at constant magnetization m . We also calculate the heat and spin modes which are purely diffusive. Explicit expressions for the viscosity, thermal conductivity, spin diffusion and thermomagnetic diffusion coefficients containing time correlation functions are derived. Our results are compared with previous ones obtained mainly within phenomenological theories.

We examine the equation of state of liquid4He at negative pressures close to the spinodal density ϱs where the hydrodynamic speed of sound vanishes. The non-analytic behavior of the equation of state and the speed of sound in the vicinity of the spinodal density are calculated in two and in three dimensions; we find for the speed of sound the non-analytic behavior mc s 2 ∼ (ϱ — ϱs)2/5 in three dimensions and mc s 2 ∼[(ϱ-ϱs)/¦ln(ϱ-ϱs)¦]1/2 in two dimensions. We then examine the low density regime numerically, using a semi-analytic microscopic theory. It is found that non-analytic exponents are visible only in a negligible density regime around the spinodal point. Estimates for the spinodal densities, and the range of critical fluctuations are provided.

The field-theoretical gauge model for a superconductor, generalized to an n/2-component complex order parameter is considered. The question of the order of the phase transition occurring in this model is discussed. Previous renormalization-group calculations suggest that in such a model a fluctuation-induced first-order phase transition occurs. We re-examine previously obtained expressions for the renormalization-group functions in the two-loop approximation in three dimensions. Special attention is being paid to the fact that the corresponding series might be the asymptotic ones and therefore have zero radius of convergence. We discuss the possible ways of analytical continuation of the series obtained. Comparing results obtained by `direct' calculations with those obtained by Pade analysis and the Pade - Borel resummation technique, the conjecture is made that in the model under consideration there still exists a possibility for a second-order phase transition.

Using Zubarev's method of nonequilibrium statistical operator, the generalized hydrodynamic equations are obtained for a model of magnetic liquid in an inhomogeneous external field. In this model the “liquid” subsystem is treated as a classical one and the “magnetic” subsystem is described by quantum mechanical methods. The properties of the transport equations are analysed in the case of a weak nonequilibrium. The equations for time correlation functions and collective mode spectrum are also found in the same manner. It is shown that the generalized hydrodynamic equations reduce to the well-known results in the limiting cases when the dynamic variables of one subsystem are formally neglected. As an illustration, a simple model of spin relaxation is considered, and the frequency matrix and the matrix of memory functions are calculated. A comparison with previous works is made.

The critical behavior near consolute points and plait points in mixtures and along lines connecting such points in the phase diagram belongs to the universality class of gas-liquid transitions in pure liquids. We give a survey of the results for the temperature dependence of transport coefficients, thermal conductivity, mass diffusion, and thermal-diffusion ratio, in mixtures within a non-asymptotic renormalization-group theory of critical dynamics. The observable critical behavior in some cases is nonuniversal and may be strongly concentration dependent. This is explained by different crossover temperatures in the singular Onsager coefficient of the order parameter and in the hydrodynamic transport coefficients. At the plait point the value of (ie1363-01) determines the crossover to the asymptotic behavior in the transport coefficients, and its smallness explains the situation in3He-4He mixtures. We also consider ionic solutions, where long-range forces may be present. The dynamical universality class in this case is different from that of mixtures with short-range interaction. As well as the “classical” static behavior for sufficient long-range interaction potentials, the dynamical critical behavior depends on the exponent of the power law for the spatial decrease in this interaction. This offers an additional possibility to determine this exponent by measuring the temperature dependence of the hydrodynamic transport coefficients.

We study the recognition properties of a modification of the Willshaw associative memory with a floating threshold analytically for the asymptotic case of large networks and single-step retrieval. The number of spurious stable states and the size of their attraction basins are calculated. Comparison is made with simulations for single- and multistep dynamics. The appropriateness of the recognition procedure based on the remanent overlap is demonstrated.

We describe a simple neural model for self-organizing feature maps based on elastic interaction between weight vectors of neurons. The model dynamics are deterministic and parallel and relax to an equilibrium configuration. The overall properties of the model depend on the elastic force used. If the force depends on the pth power of the difference of the weight vectors the density of neurons in the equilibrium configuration is given by the (1/p)th power of the signal density.

This article concludes a series of papers (R. Folk, Yu. Holovatch, and G. Moser, Phys. Rev. E 78, 041124 (2008); 78, 041125 (2008); 79, 031109 (2009)) where the tools of the field theoretical renormalization group were employed to explain and quantitatively describe different types of static and dynamic behavior in the vicinity of multicritical points. Here, we give the complete two loop calculation and analysis of the dynamic renormalization-group flow equations at the multicritical point in anisotropic antiferromagnets in an external magnetic field. We find that the time scales of the order parameters characterizing the parallel and perpendicular ordering with respect to the external field scale in the same way. This holds independent whether the Heisenberg fixed point or the biconical fixed point in statics is the stable one. The non-asymptotic analysis of the dynamic flow equations shows that due to cancelation effects the critical behavior is described - in distances from the critical point accessible to experiments - by the critical behavior qualitatively found in one loop order. Although one may conclude from the effective dynamic exponents (taking almost their one loop values) that weak scaling for the order parameter components is valid, the flow of the time scale ratios is quite different and they do not reach their asymptotic values., 22 pages, 7 figures. Revised version, references updated, Fig. 5 modified, typos corrected

Neural networks storing Hadamard patterns have been completely classified with respect to permutation symmetry. The symmetry group of the Hadamard patterns is found to be isomorphic to GL(n,F2), and the symmetry groups of the networks are explicitly constructed for the most important classes. The volumes of different equivalence classes have been calculated.

In this paper, we address the logarithmic corrections to the leading power laws that govern thermodynamic quantities as a second-order phase transition point is approached. For phase transitions of spin systems on d-dimensional lattices, such corrections appear at some marginal values of the order parameter or space dimension. We present new scaling relations for these exponents. We also consider a spin system on a scale-free network which exhibits logarithmic corrections due to the specific network properties. To this end, we analyze the phase behavior of a model with coupled order parameters on a scale-free network and extract leading and logarithmic correction-to-scaling exponents that determine its field- and temperature behavior. Although both non-trivial sets of exponents emerge from the correlations in the network structure rather than from the spin fluctuations they fulfil the respective thermodynamic scaling relations. For the scale-free networks the logarithmic corrections appear at marginal values of the node degree distribution exponent. In addition we calculate scaling functions, which also exhibit nontrivial dependence on intrinsic network properties., 15 pages, 4 figures

A complete two loop renormalization group calculation of the multicritical dynamics at a tetracritical or bicritical point in anisotropic antiferromagnets in an external magnetic field is performed. Although strong scaling for the two order parameters (OPs) perpendicular and parallel to the field is restored as found earlier, in the experimentally accessible region the effective dynamical exponents for the relaxation of the OPs remain different since their equal asymptotic values are not reached., Comment: 6 pages, 2 figures; some additions, corrected typos

We briefly review the Ising model with uncorrelated, quenched random-site or random-bond disorder, which has been controversial in both two and four dimensions. In these dimensions, the leading exponent alpha, which characterizes the specific-heat critical behaviour, vanishes and no Harris prediction for the consequences of quenched disorder can be made. In the two-dimensional case, the controversy is between the strong universality hypothesis which maintains that the leading critical exponents are the same as in the pure case and the weak universality hypothesis, which favours dilution-dependent leading critical exponents. Here the random-site version of the model is subject to a finite-size scaling analysis, paying special attention to the implications for multiplicative logarithmic corrections. The analysis is fully supportive of the scaling relations for logarithmic corrections and of the strong scaling hypothesis in the 2D case. In the four-dimensional case unusual corrections to scaling characterize the model, and the precise nature of these corrections has been debated. Progress made in determining the correct 4D scenario is outlined., Proceeding for Statistical Physics 2009 conference in Lviv, Ukraine

Spin-1/2 Ising-Heisenberg model with XYZ Heisenberg pair interaction and two different Ising quartic interactions is exactly solved with the help of the generalized star-square transformation, which establishes a precise mapping equivalence with the corresponding eight-vertex model on a square lattice generally satisfying Baxter's zero-field (symmetric) condition. The investigated model exhibits a remarkable weak-universal critical behavior with two marked wings of critical lines along which critical exponents vary continuously with the interaction parameters. Both wings of critical lines merge together at a very special quantum critical point of the infinite order, which can be characterized through diverging critical exponents. The possibility of observing reentrant phase transitions in a close vicinity of the quantum critical point is related to a relative strength of the exchange anisotropy in the XYZ Heisenberg pair interaction., 10 pages, 7 figures, presented at Statistical Physics: Modern Trends and Applications to be held on June 23-25, 2009, in Lviv, Ukraine

Dynamical scaling functions above ${T}_{c}$ for the characteristic frequencies and the dynamical correlation functions of the order parameter and the conserved density of model C are calculated in one loop order. By a proper exponentiation procedure these results can be extended in order to consider the changes in these functions using the fixed point values and exponents in two loop order. The dynamical amplitude ratio $R$ of the characteristic frequencies is generalized to the critical region. Surprisingly the decay of the shape functions at large scaled frequency does not behave as expected from applying scaling arguments. The exponent $\ensuremath{\upsilon}$ of the decay does not change when going from the critical to the hydrodynamic region although the shape functions change. The value of $\ensuremath{\upsilon}$ for the order parameter is in agreement with its value in the critical region, whereas for the conserved density it is equal to 2, the value in the hydrodynamic region.

The system of two scalar order parameters on a complex scale-free network is analyzed in the spirit of Landau theory. To add a microscopic background to the phenomenological approach, we also study a particular spin Hamiltonian that leads to coupled scalar order behavior using the mean-field approximation. Our results show that the system is characterized by either of two types of ordering: either one of the two order parameters is zero or both are nonzero but have the same value. While the critical exponents do not differ from those of a model with a single order parameter on a scale-free network, there are notable differences for the amplitude ratios and the susceptibilities. Another peculiarity of the model is that the transverse susceptibility is divergent at all $Tl{T}_{c}$, when $O(n)$ symmetry is present. This behavior is related to the appearance of Goldstone modes.

The effect of mass asymmetry on the transverse dynamics of the binary symmetric mixture is investigated by combining the molecular dynamic (MD) simulations and parameter‐free generalized collective mode (GCM) approach. The dependence of shear viscosity on the mass ratio of particles in the species is represented. A good agreement between results obtained using Green‐Kubo formula and relation for correlation time of total momentum correlation function is found in a wide range of mass asymmetry. The behaviour of two lowest lying propagating modes is studied analytically within the four‐variable approximation of GCM approach. It is established that if the mass ratio increases the propagating gap for shear waves becomes wider. However, if the concentration of heavy particles increases an opposite tendency is observed. The time correlation functions (TCFs) are calculated within this simplified four‐variable scheme and a good agreement with results of direct MD simulations for these quantities is found.

A mesoscopic description of various systems containing spherical charged particles in solvent inducing effective attraction is developed by a systematic coarse‐graining procedure. For weak ordering the theory can be reduced to the Landau‐Ginzburg or the Landau‐Brazovskii field theory, depending on the form of the effective interactions between particles. Within the framework of this theory we obtain and discuss the λ‐line and the universal sequence of phases: disordered, bcc, hexagonal, lamellar, inverted hexagonal, inverted bcc, disordered, for increasing density well below the close‐packing density. The sequence of phases agrees with experimental observations and with simulations of many self‐assembling systems. In addition to the above phases, more complex phases may appear depending on the interaction potentials. For a particular form of the short‐range attraction long‐range repulsion potential we find the bicontinuous gyroid phase (Ia3d symmetry) that may be related to a network forming cluster of colloids in a mixture of colloids and nonadsorbing polymers.

This paper studies the Yang‐Lee edge singularity of 2‐dimensional (2D) Ising model based on a quantum spin chain and transfer matrix measurements on the cylinder. Based on finite‐size scaling, the low‐lying excitation spectrum is found at the Yang‐Lee edge singularity. Based on transfer matrix techniques, the single structure constant is evaluated at the Yang‐Lee edge singularity. The results of both types of measurements are found to be fully consistent with the predictions for the (A4, A1) minimal conformal field theory, which was previously identified with this critical point.